**6. Analysis of the Results**

This section covers the discussion and conclusion on the behaviors of velocity field *U*, temperature field θ, and mass field ϕ, along with heat transfer rate ∂θ ∂*Y* , mass transfer rate ∂ϕ ∂*Y* , and skin friction ∂*U* ∂*Y* , with the variations of di fferent flow parameters. The e ffects of parameters which are taken into observation are named as magnetic field parameter, M, heat generation parameter, Q, Schmidt number, Sc, Prandtl numbers, Pr, thermophoresis parameter, Nt, and Brownian motion parameter, Nb. The obtained numerical solutions for considered governing properties are displayed in graphical form and tabulated as well. The solution detail has been split into two parts, i.e., the several locations around a sphere and in the plume region above the sphere.

### *6.1. Fluxes and Boundary Layers on the Sphere*

In this subsection, we are going to present and discuss the obtained solutions at di fferent circumferential stations around the surface of a sphere. The result demonstrated in Figure 2a–c are for velocity, temperature, and mass profiles with the variations of Schmidt number keeping the remaining parameters constant at di fferent circumferential positions of a sphere. It can be viewed that, as the Schmidt number is increased at the considered positions of a sphere, that is X = 0.1, 1.0, 2.0, and 3.0, velocity and mass profiles go down, but the opposite behavior is observed in the temperature field. In addition, it is necessary to mention that maximum magnitude for velocity distribution is achieved at position X = 1.0, but for temperature and mass concentration, it is obtained at X = 3.0. In these graphs, the simultaneous momentum and mass di ffusion convection processes have been highlighted very clearly. Figure 3a,b depicts the results for velocity, temperature field, and mass concentration corresponding to increasing values of heat generation parameter Q and the remaining parameters treated as fixed at several stations of a sphere. We can see that the temperature and mass distributions have decreasing behavior, but the opposite phenomenon is observed in the velocity distribution. One aspect which is necessary to highlight is that very minor variations are observed for temperature and velocity fields, but a reasonable change is found in mass distribution at the taken circumferential positions of a sphere. From these graphs, it is evident that the heat generation parameter balances the heat transfer mechanism in the fluid flow domain. Figure 4a–c represents the behavior of the aforesaid physical properties for di fferent values of the Prandtl number Pr. The outcomes shown in Figure 4a–c imply that, owing to the enhancement of Prandtl number at di fferent circumferential locations of a sphere, a decrease in mass and velocity distributions, but an increase in temperature distribution, are noted. It is necessary to mention that the highest magnitude for velocity is gained at circumferential points X = 1.0, while on the other hand, mass and temperature distribution secure the peak value at position X = 3.1. As the Prandtl number controls the relative thickness of the momentum and thermal boundary layer, when Pr is small, the heat di ffuses quickly as compared to the velocity. The e ffects of Brownian motion parameters on the physical properties mentioned earlier are presented in Figure 5a–c. It is noteworthy to point out that the augmentation in the Brownian motion parameter gives birth to a rise in mass distribution, but no remarkable variations are noted in the temperature and velocity fields. Figure 6a–c highlight the outcomes of profiles of velocity, temperature, and mass concentration under the action of diverse values of magnetic field parameter. It can be noticed that fluid velocity slows down as magnetic field parameter M is increased from 0.2 to 0.8 at each contemplated point around our proposed geometry and the temperature profile and mass concentration ge<sup>t</sup> smaller magnitudes for the same values of the parameters and positions. It is a point of interest that top values for flow velocity are maintained at X = 1.0 and for temperature and mass distribution, the highest values are gained at position X = 3.0. Variations in fluid velocity, temperature field, and mass concentration for increasing values of the thermophoresis parameter are demonstrated in Figure 7a–c. Very profound results are determined for all proposed properties. It is worthy to mention that the velocity of the fluid and temperature are reduced for increasing values of thermophoresis parameter Nt at the proposed positions about the surface of a sphere. On the other hand, for the same parametric conditions, mass concentration is enhanced. In Figure 8a–c, heat and mass transfer rates with skin friction are

plotted. Interestingly, it can be seen that the heat transfer rate grows well, but mass transfer and skin friction become weaker at every contemplated circumferential point of a sphere. Similar properties as discussed earlier are taken under discussion and displayed in Figure 9a–c. Benchmark results for velocity, temperature, and solutal gradients for different values of Prandtl number have been studied at various stations of a sphere. There is a reduction in skin friction and mass transfer, but an increment in heat transfer rate is noted. The results tabulated in Table 1 represents skin friction, heat transfer rate and mass transfer for varying values of Brownian motion parameter Nb. The outcomes in Table 1 imply that skin friction ge<sup>t</sup> reduced whereas heat and mass transfer rates go up as Nb is augmented at the proposed stations of a sphere. Table 2 is reflecting the influences of magnetic field parameter on aforementioned material properties. By making larger the values of magnetic field parameter all contemplated material properties ge<sup>t</sup> declined. Further, it is concluded that greatest magnitudes for skin friction, rate of heat transfer and mass transfer rate are assured at positions X = 2.0, X = 1.0, and X = 1.0, respectively. Heat generation effects on velocity gradient, heat transfer rate and mass transfer rate are illustrated in Table 3. We can claim from the displayed results that skin friction and mass transfer enhance, but the converse phenomenon occurred for the case of heat transfer. In Table 4, the impact of Schmidt number is shown. Tabulated results show that skin friction falls down, but the mass transfer rate and heat transfer rate are augmented.

**Figure 2.** Physical effects on quantities (**a**) *U*, (**b**) θ, and (**c**) ϕ versus Sc when Nt = 0.02, Nb = 0.2, Pr = 0.72, M = 0.2, Q = 1.0.

**Figure 3.** Physical effects on quantities (**a**) *U*, (**b**) θ, and (**c**) ϕ versus *Q*, when Nt = 0.02, Nb = 0.4, Pr = 0.72, M = 0.2, Sc = 1.0.

**Figure 4.** Physical effects on quantities (**a**) *U*, (**b**) θ, and (**c**) ϕ versus Pr, when Nt = 0.02, Nb = 0.4, SC = 1.0, M = 0.2, Q = 1.0.

**Figure 5.** Physical effects on quantities (**a**) *U*, (**b**) θ, and (**c**) ϕ versus Nb, when Nt = 0.02, Sc = 1.0, Pr = 7.0, M = 0.2, Q = 1.0.

**Figure 6.** Physical effects on quantities (**a**) *U*, (**b**) θ, and (**c**) ϕ versus M, when Nt = 0.02, Nb = 0.4, Pr = 7.0, Sc = 10, Q = 1.0.

**Figure 7.** Physical effects on quantities (**a**) *U*, (**b**) θ, and (**c**) ϕ versus Nt, when Sc = 1, Nb = 0.4, pr = 7.0, M = 0.2, *Q* = 1.0.

**Figure 8.** Physical effects on quantities (**a**) ∂*U*∂*Y* , (**b**) ∂θ∂*Y* , and (**c**) ∂φ∂*Y* versus, Nt when Sc = 1.0, Nb = 0.4, Pr = 7.0, M = 0.2, Q = 1.0.

**Figure 9.** Physical effects on quantities (**a**) ∂*U*∂*Y* , (**b**) ∂θ∂*Y* , and (**c**) ∂φ∂*Y* versus, Pr when Sc = 1.0, Nb = 0.4, Pr = 0.2, M = 0.2, Q = 1.0.


**Table 1.** Physical effects on quantities ∂*U*∂*Y* , ∂θ∂*Y* , and ∂φ∂*Y* versus Nb, when remaining emerging parameters are constant.

**Table 2.** Physical effects on quantities ∂*U*∂*Y* , ∂θ∂*Y* , and ∂φ∂*Y* versus M, when remaining emerging parameters are constant.


**Table 3.** Physical effects on quantities ∂*U*∂*Y* , ∂θ∂*Y* , and ∂φ∂*Y* versus *Q*, when remaining emerging parameters are constant.


**Table 4.** Physical effects on quantities ∂*U*∂*Y* , ∂θ∂*Y* , and ∂φ∂*Y* versus Sc, when remaining emerging parameters are constant.


### *6.2. Fluxes and Boundary Layers in the Plume Region*

The present subsection deals with the analysis and demonstration of the numerical solutions of the flow model developed for the case of the plume region which occurs above the sphere. Figure 10a–c illustrate the temperature profile and nanoparticles volume fraction profile for various values of Schmidt number in the plume region. All the other parametric values are fixed. We can deduce from the figures that, as Sc is augmented from 0.3 to 0.9, velocity is lowered whereas temperature and nanoparticles volume fraction profiles curves go up. It was expected that the nanoparticles volume fraction will rise corresponding to an increase in Schmidt number Sc. The influences of thermophoresis parameter *Nt*, on the material properties are highlighted in Figure 11a–c. Computed results are reflecting that velocity of the flow field gets enhanced but temperature and nanoparticles volume fraction decline as *Nt* is

increased. Graphical representations in Figure 12a–c are for the same substantial properties under the influence of several values of Brownian motion parameter Nb. It can be inferred from the displayed results that the velocity of the flow field goes down, nanoparticles volume fraction distribution goes up, but no variations are seen in the temperature field. Heat generation impacts by taking its several values on the conduct of matter properties, such as velocity, temperature, and nanoparticles volume fraction profiles, are examined in Figure 13a,b. From the sketched graphs, it is inferred that velocity and temperature field ge<sup>t</sup> larger magnitudes with the reduction in nanoparticles volume fraction by the augmentation of heat generation parameter Q. The results according to expectation satisfy the given boundary conditions and approach to the targets asymptotically. The graphs in Figure 14a–c are sketched for many values of magnetic field parameter M. It can be deduced from these plots that flow velocity and nanoparticles volume fraction rise, but no difference is found in the temperature field. The effects of various values of Prandtl number Pr on the already mentioned material properties are elaborated graphically in Figure 15a–c. We can see that velocity and temperature distributions decrease but nanoparticles volume fraction increases owing to increasing values of Pr. It was obvious that there is a reduction in field velocity and temperature profile.

**Figure 10.** Physical effects on quantities (**a**) *u*, (**b**) θ, and (**c**) ϕ versus, *Sc*, when Nt = 0.5, Nb = 0.4, Pr = 0.71, M = 0.5, Q = 0.4.

**Figure 11.** Physical effects on quantities (**a**) *u*, (**b**) θ, and (**c**) ϕ versus Nt, when *Sc* = 0.8, Nb = 0.4, Pr = 0.71, M = 0.4, Q = 0.2.

**Figure 12.** Physical effects on quantities (**a**) *u*, (**b**) θ, and (**c**) ϕ versus Nb, when Nt = 0.5, Sc = 0.3, *Pr* = 0.71, M = 0.4, Q = 0.4.

**Figure 13.** Physical effects on quantities (**a**) *u*, (**b**) θ, and (**c**) ϕ versus Q, when Nt = 0.1, Nb = 0.4, Pr = 0.71, M = 0.2, *Sc* = 0.2.

**Figure 14.** Physical effects on quantities (**a**) *u*, (**b**) θ, and (**c**) φ versus, when Nt = 0.1, Nb = 0.4, Pr = 0.71, Q = 0.2, Sc = 0.2.

**Figure 15.** Physical effects on quantities (**a**) *u*, (**b**) θ, and (**c**) φ versus Pr, when Nt = 0.1, Nb = 0.4, Q = 0.2, M = 0.2, Sc = 0.2.
